in a downhill race a skier slides down a 40 degree slope. starting from rest how far must he slide down the slope to reach a speed of 130 km/hr. ignore air friction and assume the co effiecnt of kinetic friction between skies and snow is .1
ForceNet=mg
but force net= mg*CosTheta-friction
net force= mg*cosTheta-mu*mg*SinTheta
set that equal to ma, solve for a.
Now, knowing a
Vf^2=Vi^2+2ad solve for distance d.
put velocities in m/s
To solve this problem, we can use the principles of kinematics.
Step 1: Convert the given speed from km/hr to m/s.
- 130 km/hr = 130,000 m/3600 s ≈ 36.1 m/s
Step 2: Determine the acceleration of the skier.
- Since the skier starts from rest, the initial velocity (u) is 0 m/s.
- The final velocity (v) is 36.1 m/s.
- The angle of the slope (θ) is 40 degrees.
- The coefficient of kinetic friction (μ) is 0.1.
- The acceleration (a) can be calculated using the equation: v^2 = u^2 + 2as, where s represents the distance traveled.
- Rearranging the equation, we have s = (v^2 - u^2) / (2a).
Step 3: Calculate the acceleration due to gravity.
- The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Step 4: Calculate the frictional force.
- The coefficient of kinetic friction (μ) between the skis and snow is 0.1.
- The frictional force (Ff) can be determined by multiplying the coefficient of friction by the normal force.
- The normal force (Fn) can be calculated using the equation: Fn = mg, where m is the mass of the skier.
- Since the mass is not provided, we can omit it from the calculations since it cancels out in the final result.
- Thus, we have Ff = μ * Fn = μ * m * g.
Step 5: Calculate the component of gravity along the slope.
- The component of gravity acting along the slope (Fg//) can be determined by multiplying the force of gravity (mg) by the sine of the angle of the slope.
- We have Fg// = mg * sin(θ).
Step 6: Calculate the net force.
- The net force (Fnet) is calculated as the difference between the component of gravity along the slope (Fg//) and the frictional force (Ff).
- Fnet = Fg// - Ff.
Step 7: Calculate the acceleration of the skier.
- The acceleration (a) is determined by dividing the net force (Fnet) by the mass of the skier.
- Since the mass cancels out in the final result, we can solve for a without considering the mass.
Step 8: Calculate the distance the skier slides down the slope.
- Now we can use the previously mentioned equation: s = (v^2 - u^2) / (2a).
Note: In this solution, we have omitted the effect of air friction and assumed that the coefficient of kinetic friction is constant throughout the motion.
Let's go through the calculations step-by-step.
To find the distance the skier must slide down the slope to reach a speed of 130 km/hr, we'll need to use the principles of physics and some basic trigonometry.
Let's break it down into steps:
Step 1: Convert the speed from km/hr to m/s.
To do this, we need to multiply the speed by a conversion factor. Since 1 km = 1000 m and 1 hr = 3600 s, the conversion factor is (1000 m / 3600 s). So, multiplying 130 km/hr by this conversion factor, we get:
130 km/hr * (1000 m / 3600 s) = 36.11 m/s (rounded to two decimal places).
Step 2: Calculate the component of the gravitational force parallel to the slope.
The force due to gravity can be divided into two components: one perpendicular to the slope and one parallel to the slope. The parallel component is equal to mg * sin(θ), where m is the mass of the skier (which we're assuming doesn't affect the final distance) and θ is the angle of the slope (40 degrees in this case).
Step 3: Calculate the net force acting on the skier parallel to the slope.
The net force is equal to the parallel component of the gravitational force minus the force of kinetic friction.
The force of kinetic friction can be calculated using the formula F_fric = μ * F_normal, where μ is the coefficient of kinetic friction and F_normal is equal to mg * cos(θ), where cos(θ) is the cosine of the angle of the slope.
Step 4: Calculate the acceleration of the skier.
The net force acting on the skier parallel to the slope divided by the mass of the skier will give us the acceleration. However, since we're assuming that the mass of the skier doesn't affect the final distance, we can simply consider the net force acting on the skier.
Step 5: Calculate the distance traveled.
Now we have all the necessary information to calculate the distance traveled. We can use the equation v_f^2 = v_i^2 + 2 * a * d, where v_f is the final velocity (36.11 m/s), v_i is the initial velocity (0 m/s), a is the acceleration, and d is the distance.
Rearranging the equation to solve for d, we have:
d = (v_f^2 - v_i^2) / (2 * a).
By plugging in the values we've calculated, we can find the distance.